In this paper, Lie group classification to the N-th-order nonlinear evolution equation ut=uNx + F(x1t1u1ux1, . . . 1u(N-1)x) is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensional solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group so(3) as the symmetry group of the equation, and only two realizations of sl(2, R) are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.
The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation.The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described.There are second-order nonlinear differential operators,including cubic operators and quadratic operators,which preserve an invariant subspace of submaximal dimension.A full description of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given.It is shown that the maximal dimension of invariant subspaces preserved by a second-order cubic operator is four.Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities.These solutions blow up in a finite time.
